**1. Introduction**

Many caves developed in karstic regions and basaltic regions. But ice caves, permanent ice deposit preserved within caves, are a rare geological phenomenon. An ice cave is a type of natural cave that contains significant amounts of perennial ice. The most famous of ice caves are Eisriesenwelt ice cave, Austria [1–3]; Dobšinská ice cave, Slovakia [4, 5]; Scarisoara ice cave, Romania [6, 7]; and Monlesi ice cave, Switzerland [8, 9]. The Eisriesenwelt ice cave is the largest ice cave in the world. More than 10 ice caves have been found in China, and Ningwu, Zibaishan, and Wudalianchi ice caves are the most famous ice caves.

As long ago as 1861, studies of ice caves began [10]. In the background of global climate change, seven international conferences on ice caves have been held in recent decades [11]. Several articles have documented seasonal air temperature fluctuations of several degrees from cave systems [12–14]. Consequently, in order to approximate the impact of climatic conditions on cave environments, a better interpretation of subsurface heat transfers is necessary [9]. In addition to this, ice caves are tourism resources. A better interpretation of subsurface heat transfers may assist in managing ice caves more scientifically.

Empirical calibrations have been carried out previously to evaluate the spatiotemporal distribution of cave temperature as a mathematical relation of the outside atmospheric conditions [15, 16]. In temperate karstic natural conditions, interpretation of the existence of subsurface ice deposits represents probably the most severe test for models of the magnitude and direction of heat and mass transfers induced by cave air circulation [9]. In theory and application (mathematics and engineering), the finite element method (FEM) and the finite difference method (FDM) are classical methods for finding approximate solutions of partial differential equations, which governed physical processes (including heat transfer). The details of FEM and FDM can be found in many textbooks [17, 18]. These methods would be useful for ice cave studies.

Since Ningwu ice cave was found, ice cave studies began in 1998 in China. Ningwu ice cave in Shanxi Province has been broadly reported during the past decade [19, 20], but little was known about the physical processes controlling the formation and preservation of permanent subsurface ice deposits under temperate climate conditions [21]. In addition to, basaltic cave stability is significant for ice preservation. FEM was applied to investigate the energy exchange of Ningwu ice cave and then quantitatively interpret the formation and preservation mechanism of the ice deposit. Thermal-elastic modeling was conducted to study lava tube stability during its cooling process.
